|
|
A209168
|
|
Triangle of coefficients of polynomials u(n,x) jointly generated with A209169; see the Formula section.
|
|
3
|
|
|
1, 2, 1, 4, 6, 3, 7, 16, 17, 7, 12, 37, 56, 47, 17, 20, 78, 154, 182, 128, 41, 33, 156, 378, 574, 565, 344, 99, 54, 301, 864, 1590, 1995, 1697, 915, 239, 88, 566, 1877, 4048, 6118, 6605, 4973, 2413, 577, 143, 1044, 3927, 9693, 17073, 22128, 21093
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Row n begins with F(n+2)-1, where F=A000045 (Fibonacci numbers).
Alternating row sums: 1,1,1,1,1,1,1,1,1,1,1,1,1,1,...
For a discussion and guide to related arrays, see A208510.
|
|
LINKS
|
|
|
FORMULA
|
u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.
|
|
EXAMPLE
|
First five rows:
1
2....1
4....6....3
7....16...17...7
12...37...56...47...17
First three polynomials v(n,x): 1, 2 + x, 4 + 6x + 3x^2.
|
|
MATHEMATICA
|
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|